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Donald in Mathmagic Land
Donald: Mighty strange. Huh, that's an odd-looking creature. What kind of a crazy place is this? Well, what do you know. Square roots. :Pi Creature: Pi is equal to 3.141592653589747, etc, etc, etc. :Donald: Huh? Hello? echoing :Spirit: Hello, Donald. :Donald: That's me. Where am I? :Spirit: Mathmagic Land. :Donald: Mathmagic Land? Never heard of it. :Spirit: It's the land of great adventure. :Donald: Well, who are you? :Spirit: I'm a spirit. The true spirit of adventure. :Donald: That's for me. What's next? :Spirit: A journey through the wonderland of mathematics. :Donald: Mathematics? That's for eggheads. :Spirit: Eggheads? Now, hold on, Donald. You like music, don't you? :Donald: Yeah. :Spirit: Well, without eggheads, there would be no music. :Donald: Aw. :Spirit: Come on, let's go to ancient Greece, to the time of Pythagoras, the master egghead of them all. :Donald: Pythagoras? :Spirit: The father of mathematics and music. :Donald: offscreen Mathematics and music? :Spirit: Ah, you'll find mathematics in the darnedest places. Watch. First, we'll need a string. :Donald: Hey! :Spirit: Stretch it good and tight. Plunk it. :tone :Spirit: Now divide in half. Plunk again. :tone :Spirit: You see, it's the same tone one octave higher. Now, divide the next section. :tone :Spirit: And the next. :tone :Spirit: Pythagoras discovered the octave had a ratio of two to one. With simple fractions, he got this: :scale of tones :Spirit: And from this harmony in numbers, developed the musical scale of today. :scale :twangs :Donald: By golly, you do find mathematics in the darndest places. :Spirit: You can imagine how excited Pythagoras was when he shared his findings with his pals, a faternity of eggheads, known as the Pythagoreans. whispers They used to meet in secret to discuss their mathematical discoveries. Only members were allowed to attend. They had a secret emblem, the pentagram. Let's see what the topic is for today. :musical scale :playing scale :off-key note :off-key note :correct note :musical flourish :music together :Donald: What's going on? :Spirit: Shh, it's a jam session. :Donald: Give me something with a beat. :Spirit: Shh. :drumming :jazz :Spirit: So from these eggheads, the Pythagoreans, with their mathematical formula came the basis of our music of today. :music :music :Donald: Pythag, old boy, put 'er there. Well, I'll be a gosh darn egghead. chuckles :Spirit: It was our old friend Pythagoras who discovered that the pentagram was full of mathmagic. The two shorter lines combined exactly equal the third. And this line shows the magic proportions of the famous golden section. The second and third lines exactly equal the fourth. Once again, we have the golden section. But this is only the beginning. Hidden within the pentagram is a secret for creating a golden rectangle, which the Greeks admired for its beautiful proportions and magic qualities. The star contains the golden rectangle many times over. It's a most remarkable shape. It can mathematically reproduce itself indefinitely. All these rectangles have exactly the same proportions. This figure also contains a magic spiral that repeats the proportions of the golden section into infinity. To the Greeks, the golden rectangle represented a mathematical law of beauty. We find it in their classical architecture. The Parthenon, perhaps one of the most famous of early Greek buildings, contains many golden rectangles. These same golden proportions are also found in their sculpture. In the centuries that followed, the golden rectangle domintated the idea of beauty and architecture throughout the western world. The cathedral of Notre Dame is an outstanding example. The renaissance painters knew this secret well. Today, the golden rectangle is very much a part of our modern world. Modern painters have rediscovered the magic of these proportions. Indeed, this ideal proportion is to be found in life itself. :Donald: offscreen Boy, oh, boy, oh, boy. This is mathematics? I like mathematical figures like that. :Spirit: Ah, ah, ah, Donald. :Donald: Let me try it. :Spirit: No, no. :Donald: Ideal proportion. :Spirit: Not quite. Uh-uh. No, I'm afraid not. Well, we can't all be mathematically perfect. :Donald: Oh, yeah? There. I knew I could do it. :Spirit: Now that you're all pent up in a pentagon, let's see how nature uses this same mathematical form. The petunia. The star jasmine. The starfish. The wax flower. There are literally thousand of members in good standing in nature's Pythagorean society of the star. All nature's works have a mathematical logic, and her patterns are limitless. The magic proportions of the golden section are often found in the spirals of nature's designs. The profusion of mathematical forms brings to mind the words of Pythagoras, "Everything is arranged according to number and mathematical shape." Yes, there is mathematics in music, in art, in just about everything. And as the Greeks had guessed, the rules are always the same. Well, Donald, did you enjoy your geometrical journey? :Donald: Gee, Mr. Spirit, there's a lot more to mathematics than two times two. :Spirit: That's right, Donald, and you can find mathematics in games too. :Donald: Games? Oh, boy. :Spirit: Let's begin with a game that's played on squares. :Donald: Checkers? :Spirit: No, chess. :Donald: Chess? :Spirit: A mathematical contest between two minds. It's a game that has been enjoyed for centuries by kings and commoners. In fact, Lewis Carroll, a famous mathematician with a literary mind, used chess as a setting for classic tale, Through the Looking-Glass. Alice found herself face to face with a none too friendly group of chess pieces. :Chess Queen: Good heavens, what's this? :Chess King: Upon my soul, it appears to be a lost pawn. :Donald: I'm no pawn. I'm Donald Duck. :Chess Queen: He says he's Donald Duck. :Chess King: Preposterous. :Chess Queen: Or it could be an Alice. :Donald: Alice? :Chess King: No, no, no, it's a lost pawn. :Chess Queen: Lost pawn! Stop that pawn! :Donald: Help, Mr. Spirit! Help! Help! Help! yelling sighs Phew! That was close. :Spirit: Now you can look at this game from a safer perspective. Chess is a game of calculated strategy. And since the board is geometrical, the moves are mathematical. Checkmate. And the game is over. :Donald: That's very interesting. What's next? :Spirit: Practically all games are played on geometrical areas. The baseball field is a diamond. :Donald: Oh, boy! :Spirit: And without mathematics we couldn't even keep score. Football is played on a rectangle divided by yard lines. :grunts :Spirit: Basketball is a game of circles, spheres and rectangles. Even hopscotch has its multiple squares. :Donald: What's next? Tiddlywinks? :Spirit: No. A mathematical game played on a field of two perfect squares, using three perfect spheres and a lot of diamonds. In other words, billiards. :Donald: Oh, boy! That's for me. :Spirit: You know the game, don't you, Donald? :Donald: Of course. The cue ball has to hit the other two balls like this. :Spirit: Now, let's see how an expert at three-cushion billiards uses his head. :Donald: offscreen Three-cushion? :Spirit: Yes, the cue ball not only has to hit both the other balls, but it must contact at least three cushions before it hits the final ball. One, two, three. One, two, three. It takes an expert to make several shots in succesion. One, two, three, four, five, six. :Donald: Wow! That was a lucky shot. :Spirit: Luck? No. It's skill. For this game, you have to know all the angles. :Donald: offscreen One, two, three, four, five, six, seven. That's amazing! How does he do it? :Spirit: First, there's technique. He's striking the cue ball low so it'll spin backwards. Hitting the ball on the right side will make it hug the rail. These trick shots take a lot of practice. :Donald: chuckling He missed it! chuckling :Spirit: One, two, three. :Donald: What's so mathematical about that? :Spirit: Oh, this game takes precise calculation. He figures out each shot in his head. He could play it like this but it calls for quite a bit of luck. There is a better choice. For this, he uses the diamond markings on the rail as a mathematical guide. First, he figures the natural angle for hitting the object balls. And then he finds that his cue ball must bounce off the number-three diamond. Next, he gets ready for the shot and he needs a number for his cue positon. This calls for a different set of numbers. :Donald: offscreen Very confusing, isn't it? :Spirit: Not when you got the hang of it. You see, the cue position is four. Now, a simple subtraction. Three from four is one. So if he shoots for the first diamond he should make it. It's called "playing the diamond system." Natural angle two. Cue position, one and a half, two, two and a half, three, three and a half. Two from three and a half is one and a half. So shoot halfway between the first and second diamonds. :Donald: There's nothing to it. Let me try. Let's see now. It I hit it here, it'll bounce there and... No, here. If I hit it here, four and a half minus three, three and a half plus four added to two, uh, divided into... I guess I should shoot about here. :Spirit: No, no, Donald, there's no guesswork to mathematics. It's quite simple. Natural angle for the hit: two. Cue position: three and a half. How much is three and a half minus two? :Donald: offscreen Um, one and a half. Hey, it works! Oh, boy! It's a cinch! If I hit it here, uh, three and a half plus four to four and a half minus three... mumbling :Spirit: You're making it tough for yourself, Donald. :Donald: How do you like that for mathematics, Mr. Spirit? :Spirit: Wonderful, Donald. And now, you're ready for the most exciting game of all. :Donald: Oh, boy! :Spirit: And the playing field for this game is in the mind. Oh oh. Look at the condition of your mind. Antiquated ideas, bungling, false concepts, superstitions, confusion. To think straight we'll have to clean house. :sweeping :Spirit: There. That's more like it. A nice clean sweep. This game is played with circles and triangles. Think of a perfect circle. A perfect circle. Perfect circle. Perfect. Ahhh. Put a triangle inside and turn it. Now spin the circle and what have you got? :Donald: offscreen A ball! :Spirit: Yes, a sphere, the shape of things is first discovered in the mind. Slice off the top and we have a... :Donald: offscreen A magnifying glass. :Spirit: That's right. A lens is a section of a sphere. All optical instruments are created through mathematics. You see, there's a lot more to mathematics than just numbers and equations. Let's get back to our circle and triangle. Roll it and we have a... :Donald: offscreen A wheel. :Spirit: The circle has been the basis for many of man's important inventions. The mind can create the most amazing things. If we spin the triangle we have a... :Donald: offscreen Cone. :Spirit: Slice the cone. :chuckles :Spirit: The cone is full of useful mathematical shapes. Slice it again. Slice it several times. The orbits of all planets and satellites can be found in the cone. No matter how you slice it, it's always mathematics. A slice like this gives us the reflector of a searchlight. A slice like this, the mirror of a giant telescope. A line on a cone and we have a drill. And a spring. Now you're ticking. :ticking :dialing :Woman: Number please. :band music plays :drumming :crashing :Spirit: The mind is the birthplace for all of man's scientific achievements. The mind knows no limits when used properly. Think of a pentagram, Donald. Now, put another inside, a third and a fourth. No pencil is sharp enough to draw as fine as you can think, and no paper large enough to hold your imagination. In fact, it is only in the mind that we can conceive infinity. Mathematical thinking has opened the doors to the exciting adventures of science. :Donald: I'll be doggoned. I've never seen so many doors before. :Spirit: Each discovery leads to many others, an endless chain. :Donald: Hey! Hey! What's the matter with these doors? Hey! These doors won't open. They're locked. :Spirit: Of course they are locked. These are the doors of the future and the key is... :Donald: Mathematics! :Spirit: Right, mathematics. The boundless treasures of science are locked behind those doors. In time, they will be opened by the curious and inquiring minds of future generations. In the words of Galileo, "Mathematics is the alphabet with which God has written the universe." Category:Disney